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The Massachusetts unemployment rate last month fell below 7 percent for the first time in three years, but employers cut jobs - a sign that the state’s economy may be slowing after a burst of growth early last year.

Economists said a slowdown, or rather a leveling off, is to be expected after the state enjoyed some of the most robust job gains in years during the first half of 2011.

“The growth rate was so strong in the first half of the year - that may account, partly, for why the growth was so slow in the second half,’’ said Alan Clayton-Matthews, a Northeastern University economics professor. “If you were hiring like crazy, then you filled the spots you needed to fill.’’

In Massachusetts, the jobless rate fell to 6.8 percent in December, from 7 percent in the previous month, the state’s Executive Office of Labor and Workforce Development reported yesterday. It was the lowest unemployment rate since December 2008.

Employers, however, cut more than 6,000 jobs last month, and revised data showed they added far fewer jobs in November than first reported - just 300 compared with initial estimates of 5,000.

The unemployment rate and job statistics are based on separate surveys, and sometimes the results diverge. The unemployment rate is estimated from a survey of households, and job growth from a survey of employers.

Only four of 10 private employment sectors added jobs in Massachusetts last month, with the largest increases in trade, transportation, and utilities, which gained 2,300 jobs. Manufacturing added 1,100 jobs.

The additions were offset by losses in key sectors that help drive the Massachusetts economy. Professional, scientific, and business services - which includes a variety of technology firms - lost 5,000 jobs in December. Employment in education and health fell by 2,400 jobs, and leisure and hospitality lost 1,200 jobs.

On the bright side, Massachusetts added nearly 41,000 jobs in 2011, helping to drop the state’s unemployment rate 1.5 percentage points from 8.3 percent in January, and 2 points from its 8.8 percent peak in October 2009. The state’s unemployment rate also remains well below the national average, which was 8.5 percent in December.

While the Massachusetts economy has grown at a faster rate than the United States as a whole, the nation now appears to be catching up as the national recovery gains momentum. US employers added 200,000 jobs in December and the national unemployment rate has plunged more than a half-point since June.

Yesterday, the Labor Department reported that first time claims for unemployment benefits fell by 50,000 last week to 352,000 the lowest level since April 2008.

In 2011, payroll employment in Massachusetts grew at a 1.3 percent rate, the same rate as job growth nationally. But in the first half of the year, Massachusetts employment grew at an annual rate of about 2.5 percent, double the national rate. In the last six months of 2011, however, job growth in the state was essentially flat, while employers nationally added jobs at a 1.4 percent annual rate.

Brian Bethune, an economics professor at Amherst College, said the local job market is being held back by a lack of growth in the number of small businesses in the state.

“We have a situation where there are not a lot of new start-ups, not a lot of new companies being created,’’ he said. “While Massachusetts is doing pretty well and the unemployment rate certainly looks encouraging, there’s not a lot of dynamism in the economy.’’

Clayton-Matthews said December’s declines are not too worrisome given the state’s growth in early 2011, but he is concerned about looming challenges: For instance, debt problems in Europe, a major market for technology, drugs, and other Massachusetts products, could “put a damper on 2012.’’

“There are some real risks with the situation in Europe, especially for Massachusetts,’’ he said. “We’re heavily concentrated in Europe as a destination for our exports.’’

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